Problem 27 - Algebra and Trigonometry

Question

verify each identity. $$ \frac{\sin x-\sin y}{\sin x+\sin y}=\tan \frac{x-y}{2} \cot \frac{x+y}{2} $$

Step-by-Step Solution

Answer

After transforming both the left-hand side and right-hand side of the identity using relevant trigonometric identities, it's observed that both sides equal to \( \frac{\tan x - \tan y}{\tan x + \tan y} \). Thus, the given identity is verified.

Step 1 - Express sine function in terms of sum and difference of two angles

Rewrite the left hand side of the equation using the formula for sine of sum and difference. Remember that \( \sin(x-y) = \sin x \cos y - \cos x \sin y\) and \( \sin(x+y) = \sin x \cos y + \cos x \sin y\). So the left hand side of the equation becomes: \[ \frac{\sin x \cos y - \cos x \sin y}{\sin x \cos y + \cos x \sin y} \]

Step 2 - Simplify the expressions

Divide numerator and denominator of the expression obtained in step 1 by \( \cos x \cos y \), we get \[ \frac{\tan x - \tan y}{\tan x + \tan y} \].

Step 3 - Express tangent in terms of half angles

Now, rewrite the right hand side of the original equation using half angle formulas. For any angle \( a \), \[ \tan a = \frac{1-\cos2a}{\sin2a} \]. So the right hand side of the original equation can be expressed as \[ \frac{1-\cos(x-y)}{\sin(x-y)} \cdot \frac{\sin(x+y)}{1-\cos(x+y)} \]. Apply the formula for sine and cosine of sums, it can be simplified to \[ \frac{\tan x - \tan y}{ \tan x + \tan y} \] which now equals to the left hand side.

Step 4 - Identity verified

Having the right hand side of the original equation transformed and matched the left hand side, the identity is thus verified.

Key Concepts

Sine Sum and Difference Formulas

The sine sum and difference formulas are essential tools in trigonometry for breaking down complex angles into more manageable ones. For example, the sine of the sum of two angles can be expressed as
\( \text{sin}(x + y) = \text{sin} x \text{cos} y + \text{cos} x \text{sin} y \).
Similarly, for the difference of two angles, the formula is
\( \text{sin}(x - y) = \text{sin} x \text{cos} y - \text{cos} x \text{sin} y \).
Using these formulas, we can transform expressions involving \text{sin} x \text{and} \text{sin} y into products and differences, simplifying the process of solving trigonometric identities. Moreover, these formulas underpin many other trigonometric relationships, making them foundational for understanding further concepts in trigonometry.

Tangent Function

The tangent function, denoted as \text{tan}, is another fundamental trigonometric function defined as the ratio of the sine and cosine of an angle:
\( \text{tan} x = \frac{\text{sin} x}{\text{cos} x} \).
In the context of identity verification, understanding how the tangent function behaves with respect to the sum and difference of angles can greatly simplify the process. By dividing expressions by \text{cos} x \text{cos} y, as shown in our step-by-step solution, the tangent function allows us to move from an equation involving sines and cosines to one with tangents, often leading to a more straightforward verification.

Half Angle Formulas

Half angle formulas are powerful when dealing with trigonometric expressions involving half angles. These formulas express the sine, cosine, or tangent of half an angle in terms of the trigonometric functions of the original angle. For instance, the half angle formula for tangent is
\( \text{tan} \frac{x}{2} = \frac{1-\text{cos} x}{\text{sin} x} \),
or alternatively,
\( \text{tan} \frac{x}{2} = \frac{\text{sin} x}{1+\text{cos} x} \).
When you come across an expression like \tan \frac{x-y}{2} \text{or} \tan \frac{x+y}{2} in an identity, using the half-angle formulas can help you express tangents in terms of the sine and cosine of full angles, making the verification of an identity more accessible.

Identity Verification

Identity verification in trigonometry involves proving that two different looking trigonometric expressions are actually the same. To verify an identity means to show that, through a series of algebraic manipulations and trigonometric substitutions, one side of an equation can be transformed to match the other side. The goal is to reach a common expression, which confirms the identity.
This process often involves using a combination of trigonometric identities, including sum and difference formulas, half angle formulas, and properties of fundamental trigonometric functions like the tangent. Patience and practice in manipulating these expressions are key to becoming proficient in verifying trigonometric identities.